Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

Let C be a nonempty closed convex subset of a Banach space E with the dual E*, let T : C -> E* be a Lipschitz continuous mapping and let S : C -> C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T - f, C)): find x is an element of C such that < y - x, Tx - f > >= 0, for all y is an element of C, where f is an element of E* is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J. L. Li's (J. Math. Anal. Appl. 295: 115 - 126, 2004) iterative algorithm for finding approximate solutions of VI(T - f, C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI(T - f, C), which is also a fixed point of S.